American Put Exercise Boundaries and Stopping Regions
Summary
The document explains how an American put’s early-exercise boundary divides stock-price and time states into stopping and continuation regions. At or below the boundary, exercising yields the immediate payoff; above it, holding the option is more valuable than exercising at once. The boundary is described as independent of the current stock price, nondecreasing over time, and equal to the strike at maturity.
The answer clarifies the shape of the boundary: before expiration it lies below the strike, because exercising early for a zero payoff would not be optimal. It describes a figure in which the boundary rises toward the strike, with the continuation region above it and the stopping region below. The text provides a qualitative correction to the question’s proposed plot, but no plotted data, derivation, or parameter-specific boundary. The geometry is a conceptual description for a finite-maturity American put, not a numerical exercise policy for a particular market or model.
Key ideas
- The exercise boundary separates immediate exercise states from continuation states.
- A put should be exercised when the stock price is at or below the boundary.
- Before maturity, the boundary lies below the strike; at maturity it reaches the strike.
- The described stopping region is below the boundary, while the continuation region is above it.
- The document gives a qualitative shape rather than a calibrated boundary.
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Full text
# Figure of Stopping and Continuation Region
# Figure of Stopping and Continuation Region
I am reading Alternative Characterizations of American Put Options by Carr et al. It is stated there that:
> Consider an American put option on the stock with strike price $K$ and maturity date $T$. Let $P_t$ denote the value of the American put at time $t \in [0,T]$. For each time $t \in [0,T]$, there exists a critical stock price, $B_t$ , below which the American put should be exercised early, i.e., $$if S_t \leq B_t, then P_t =max[0,K-S_t]\tag{2}$$ $$if S_t > B_t, then P_t >max[0,K-S_t]\tag{3}$$ The exercise boundary is the time path of critical stock prices, $B_t$ ,$t \in [0,T]$. This boundary is independent of the current stock price $S_0$ and is a smooth, nondecreasing function of time t terminating in the strike price, i.e. $B_T = K$. The put value is also a function, denoted $P(S,t)$, mapping its domain $D ≡ (S,t) ∈ [0,\infty)×[0,T]$ into the nonnegative real line. The exercise boundary,$B_t$ ,$t \in [0,T]$, divides this domain $D$ into a stopping region $S ≡ [0,B_t ]×[0,T]$ and a continuation region $C ≡ (B_t ,∞)×[0,T]$ (see Figure 1).
However, there is no Figure 1 in that article. I wonder what is the figure. In my opinion, the figure is like below
Is it correct? If it is not, then what is the correct figure? Thanks
## Answer by LocalVolatility (score 6, accepted)
https://quant.stackexchange.com/a/30081
The exercise boundary $B_t$ for a finite maturity American put option is not a constant function of time as in your plot. As mentioned in the excerpt, $B_T = K$ at maturity. But for $t < T$, we have $B_t < K$ as you would never pre-maturely exercise to receive a zero payoff.
Below is a plot of the early exercise boundary that I once produced for a lecture. Note that I used a slightly different notation $E$ is the exercise price (the red line). The optimal exercise boundary is $S_t^*$ (the blue curve). The continuation region is in white and the stopping region in grey.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.